Toan T. Nguyen

Remarks on the ill-posedness of the Prandtl equation

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some

Spectral stability of Prandtl boundary layers: An overview

C^\infty

Stability of radiative shock profiles for hyperbolic–elliptic coupled systems

initial data, local in time

Spectral instability of characteristic boundary layer flows

C^\infty

STABILITY OF SCALAR RADIATIVE SHOCK PROFILES

solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.

SPECTRAL STABILITY OF NONCHARACTERISTIC ISENTROPIC NAVIER-STOKES BOUNDARY LAYERS

Abstract In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.

On the Kinetic Equation in Zakharov's Wave Turbulence Theory for Capillary Waves

Abstract Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small amplitude shock profiles of general systems of coupled hyperbolic–elliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler–Poisson model for a radiating gas. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator, with the main difficulty being the construction of the resolvent kernel in the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in standard fashion through linear estimates derived from these pointwise bounds, combined with energy estimates of nonlinear damping type.

Towards nonlinear stability of sources via a modified Burgers equation

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number:

Ill-Posedness of the Hydrostatic Euler and Singular Vlasov Equations

R \to \infty

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of